Division (mathematics)

About: Division (mathematics) is a research topic. Over the lifetime, 12717 publications have been published within this topic receiving 87814 citations.

Fair Division of Indivisible Goods: A Survey

Abstract: Allocating resources to individuals in a fair manner has been a topic of interest since the ancient times, with most of the early rigorous mathematical work on the problem focusing on infinitely divisible resources. Recently, there has been a surge of papers studying computational questions regarding various different notions of fairness for the indivisible case, like maximin share fairness (MMS) and envy-freeness up to any good (EFX). We survey the most important results in the discrete fair division literature, focusing on the case of additive valuation functions and paying particular attention to the progress made in the last 10 years.

On finding an envy-free Pareto-optimal division

Abstract: This paper describes an algorithm to find an (α-)envy-free Pareto-optimal division in the case of a finite number of homogeneous infinitely divisible goods and linear utility functions. It is used to find an allocation in the classical cake division problem that is almost Pareto-optimal and α-envy-free.

SEERAD: A high speed yet energy-efficient rounding-based approximate divider

Abstract: In this paper, a high speed yet energy-efficient approximate divider for error resilient applications is proposed. For the division operation, the divisor is rounded to a value with a specific form resulting in the transformation of the division operation to the multiplication one. The proposed approximate divider enjoys the flexibility of increasing the accuracy at the price of higher delay and hardware usage. The efficacy of the proposed approximate divider is evaluated in comparison to three different implementations of the SRT divider. The results show that the delay and energy consumption of the proposed approximate divider are, on average, 14 and 300 times smaller than those of the Radix-2 SRT with the carry-save reminder computation. Additionally, the effectiveness of the proposed approximate divider is studied in an image division operation performed in image processing applications. The results suggest the appropriateness of the proposed approximate divider for digital signal processing applications.

Compatible hardware for division and square root

Abstract: Hardware for radix four division and radix two square root is shared in a processor designed to implement the proposed IEEE floating-point standard. The division hardware looks ahead to find the next quotient digit in parallel with the next partial remainder. An 8-bit ALU estimates the next remainder's leading bits. The quotient digit look-up table is addressed with a truncation of the estimate rather than a truncation of the full partial remainder. The estimation ALU and the look-up table are asymmetric for positive and negative remainders. This asymmetry reduces the width of the ALU and the number of minterms in the logic equations for thy look-up table. The square root algorithm obtains the correctly rounded result in about two division times using small extensions to the division hardware.

Encrypted integer division

Abstract: When processing signals in the encrypted domain, homomorphic encryption can be used to enable linear operations on encrypted data Integer division of encrypted data however requires an additional protocol with the server and will be relatively expensive We present new solutions for dividing encrypted data, having low computational complexity Two protocols for computing exact division, and two for approximating the division result